All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$44.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$15.00$ each for teachers and $$8.50$ each for students, and the group paid $$119.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 44.5}$ ${15x+8.5y = 119.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-15x-10.5y = -133.5}$ ${15x+8.5y = 119.5}$ Add the top and bottom equations together. $ -2y = -14 $ $ y = \dfrac{-14}{-2}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {5x+3.5y = 44.5}$ to find $x$ ${5x + 3.5}{(7)}{= 44.5}$ $5x+24.5 = 44.5$ $5x = 20$ $x = \dfrac{20}{5}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {15x+8.5y = 119.5}$ and get the same answer for $x$ ${15x + 8.5}{(7)}{= 119.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.